Find the inverse of the following matrix using transformation method.

Engineering

Mathematics

Algebra of Matrices

Inverse of a Matrix

Question

Find the inverse of the following matrix using transformation method.
$[\begin{array}{cc} 2 & - 3 \\ - 1 & 2 \end{array}]$

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Let $A = [\begin{array}{cc} 2 & - 3 \\ - 1 & 2 \end{array}]$

C = [A | I]

= [\begin{array}{cc} 2 & - 3 \\ - 1 & 2 \end{array} ∣ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}]

R₁ → R₂ + R₁

[\begin{array}{cc} 1 & - 1 \\ 1 & 2 \end{array} ∣ \begin{array}{cc} - 1 & 2 \\ 0 & 1 \end{array}]

R₂ → R₂ + R₁

[\begin{array}{cc} 1 & - 1 \\ 0 & 1 \end{array} ∣ \begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}]

R₁ → R₂ + R₁

[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} ∣ \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array}]

Hence,

A^{- 1} = [\begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array}]